By Lluís Bruna,Joan Girbau

V ? V ?K? , three 2 2 R ? /?x ok i i g V T G g ?T , ? G g g four ? R ? ? G ? T g g ? h h ? 2 2 2 2 ? ? ? ? ? ? ? h ?S , ?? ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 three S T S T? T?. ? ˜ T S 2 2 2 2 ? ? ? ? ? ? ? h . ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 three g h h ?? g T T g vacuum M n R n R Acknowledgements n R bankruptcy I Pseudo-Riemannian Manifolds I.1 Connections M C n X M C M F M C X M F M connection covariant spinoff M ? X M ×X M ?? X M X,Y ?? Y X ? Y ? Y ? Y X +X X X 1 2 1 2 ? Y Y ? Y ? Y X 1 2 X 1 X 2 ? Y f? Y f?F M fX X ? fY X f Y f? Y f?F M X X ? torsion ? Y?? X X,Y X,Y?X M . X Y localization precept Theorem I.1. permit X, Y, X , Y be C vector ?elds on M.Let U be an open set

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